Midpoints of sides may be defined in terms of reflections in points in hyperbolic geometry. Reflections are defined by 2x2 trace zero matrices associated to points. The case of a reflection in a null point is somewhat special. The crucial property of reflection is that it preserves perpendicularity, which then implies that reflections send lines to lines. Midpoints of a side bc can be constructed with a straightedge when they exist, and in general there are two of them! This is a big difference with Euclidean geometry. Bisectors of vertices are defined by duality.CONTENT SUMMARY: pg 1: @00:10 point/matrix correspondence; reflection matrix conjugation theorem; exercise 16.1pg 2: @03:28 Definition of reflection of a general pointpg 3: @07:32 another example; Null reflection theorem; proof (exercise 16-2)pg? 4: @09:29 Matrix perpendicularity theorem; reflections as generators of isometries in hyperbolic geometrypg 5: @14:30 Reflection (preserves) perpendicularity theorem; remark about trace; proofpg 6: @18:11 reflection (preserves) lines theorem; proof; Line/point reflection notationpg 7: @21:24 exercise 16-3; Concept of Midpoint between 2 pointspg 8: @24:26 Geometrical construction concerning midpointspg 9: @28:59 Another geometrical construction concerning midpoints; Harmonic quadrangle and harmonic conjugates UHG2 revisitedpg 10: @31:42 another midpoints constructionpg 11: @33:34 Not? all sides have midpoints; side/vertex midpoints/bisectors (THANKS to EmptySpaceEnterprise)
Questions about UnivHypGeom16: Midpoints and bisectors
Want more info about UnivHypGeom16: Midpoints and bisectors?
Get free advice from education experts and Noodle community members.